# The benches of a gallery in a cricket stadium are 1 m in wide and 1 m high.a batsman strikes the ball at a level one meter above the ground and hits a mammoth sixer .The ball starts at 35 m/s at an angle of 53 with the horizontal.the benches are perpendicular to the plane of motion and the first bench is 110 m from the batsman .On which bench will the ball hit?

1
by Souvik118

2014-09-25T17:13:40+05:30

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See diagram.  The ball is bit at x =0 y =1 m.

slope of stair case of benches
=  increase in y / increase in x    =  height of step / width of step

The line joining the feet of the benches is =  (y - 1) / (x - 110 )  = 1 / 1
y = x - 109     - - equation 1

Equation of trajectory of cricket ball is :
x = u cos Ф t
t = x / u cos Ф

y  = u sinФ t - 1/2  g  t²
= x tan Ф -  g x² / ( 2 u² cos² Ф )
y  =   1.327 x  -   9.8 * x² / (2 *35² * 0.362 )
=  1.327 x  -  0.011 x²    ---- equ 2

substitute y from equation  1

x - 109 =1.327 x - 0.011 x²

0.011 x² - 0.327 x - 109 = 0

Δ = 4.9
x =  (0.327 +- 2.21 )/2*0.011

x =  115.32  meters

y at 115.32 m   =  115.32 - 109 = 6.32 meters

So it is the 6th bench,  as 1st bench is from 110m to 111m,   6 th bench is from 115 m to 116m.

Height y  Back rest of the First bench is from  1 to 2 m  ,
back rest of the sixth bench is 6 to 7 meters high

6th bench

but i think values of x are coming imaginary
ok plz let me check the answer once